QUASI m-CAYLEY STRONGLY REGULAR GRAPHS

Abstract

Abstract. We introduce a new class of graphs, called quasi m-Cayley graphs, having good symmetry properties, in the sense that they admit a group of automorphisms G that fixes a vertex of the graph and acts semiregularly on the other vertices. We determine when these graphs are strongly regular, and this leads us to define a new algebro-combinatorial structure, called quasi-partial difference family, or QPDF for short. We give several infinite families and sporadic examples of QPDFs. We also study several properties of QPDFs and determine, under several conditions, the form of the parameters of QPDFs when the group G is cyclic.

Quasi m-Cayley circulants

“…Let Γ be the Paley graph P (p), where p is a prime, such that p ≡ 1 (mod 4). It is well known that the Paley graphs are connected arc-transitive circulants, and, as was observed in [7], they are also strongly quasi 2-Cayley graphs.…”

“…In this paper we consider quasi-semiregular actions on graphs, a natural generalization of semiregular actions on graphs, which have been an active topic of research in the last decades (see, for example, [1,2,3,4,5,8,9,11]). Following [7] we say that a group G acts quasi-semiregularly on a set X if there exists an element ∞ in X such that G fixes ∞, and the stabilizer G x of any element x ∈ X\{∞} is trivial. The element ∞ is called the point at infinity.…”

“…Quasi m-Cayley graphs were first defined in 2011 by Kutnar, Malnič, Martinez and Marušič [7], who showed which strongly quasi m−Cayley graphs are strongly regular graphs.…”

Eigenvalues of Cayley graphs

“…In [190], Kutnar et al introduced a new class of graphs, called quasi n-Cayley graphs. Such graphs admit a group of automorphisms that fixes one vertex of the graph and acts semiregularly on the set of the rest vertices.…”

Symmetric graphs of valency 4 having a quasi-semiregular automorphism

“…As a matter of convenience, we exclude the identity permutation as a quasi-semiregular permutation, which follows from [11,43] but not from [20,25]. Quasi-semiregular permutation groups were first introduced in 2013 by Kutnar, Malnič, Martínez, Marušič [25], where they initiated the study of graphs with a quasi-semiregular automorphism group, and later, Hujdurović [20] classified circulants which have a quasi-semiregular automorphism group with at most 5 orbits. There are a large class of vertex-transitive graphs admitting quasi-semiregular automorphisms which are called GFRs.…”